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Groupoid

A universal algebra with one binary operation. It is the broadest class of such algebras: groups, semi-groups, quasi-groups — all these are groupoids of a special type. An important concept in the theory of groupoids is that of isotopy of operations. On a set $G$ let there be defined two binary operations, denoted by $(\cdot)$ and $(\circ)$; they are isotopic if there exist three one-to-one mappings $\alpha$, $\beta$ and $\gamma$ of $G$ onto itself such that $a\cdot b=\gamma^{-1}(\alpha a\circ\beta b)$ for all $a,b\in G$ (cf. Isotopy (in algebra)). A groupoid that is isotopic to a quasi-group is itself a quasi-group; a groupoid with a unit element that is isotopic to a group, is also isomorphic to this group. For this reason, in group theory the concept of isotopy is not used: For groups isotopy and isomorphism coincide.

A groupoid with cancellation is a groupoid in which either of the equations $ab=ac$, $ba=ca$ implies $b=c$, where $a$, $b$ and $c$ are elements of the groupoid. Any groupoid with cancellation is imbeddable into a quasi-group. A homomorphic image of a quasi-group is a groupoid with division, that is, a groupoid in which the equations $ax=b$ and $ya=b$ are solvable (but do not necessarily have unique solutions).

A set with one partial binary operation (i.e. one not defined for all pairs of elements) is said to be a partial groupoid. Any partial subgroupoid of a free partial groupoid is free.

References

O. Boruvka, "Foundations of the theory of groupoids and groups" , Wiley (1976) (Translated from German)

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R.H. Bruck, "A survey of binary systems" , Springer (1958)

Comments

There is another, conflicting, use of the term "groupoid" in mathematics, which was introduced by H. Brandt [a1]. A groupoid may conveniently be defined as a (small) category in which every morphism is an isomorphism; equivalently, it is a set $G$ equipped with a unary operation $g\mapsto g^{-1}$ and a partial binary operation $(g,h)\mapsto gh$ satisfying

1) $gg^{-1}$ and $g^{-1}g$ are always defined;

2) $gh$ is defined if and only if $g^{-1}g=hh^{-1}$;

3) if $gh$ and $hk$ are defined, then $(gh)k$ and $g(hk)$ are defined and equal;

4) each of $g^{-1}gh$, $hg^{-1}g$, $gg^{-1}h$, and $hgg^{-1}$ is equal to $h$ if it is defined.

Groupoids, as a special case of categories, play an important role in many areas of application of category theory, including algebra [a2], different geometry [a3] and topology [a4], [a5].